haftmann@29655: (*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@15300:     Copyright   1996  University of Cambridge
paulson@15300: *)
paulson@15300: 
paulson@15300: header {* Equivalence Relations in Higher-Order Set Theory *}
paulson@15300: 
paulson@15300: theory Equiv_Relations
haftmann@51112: imports Big_Operators Relation
paulson@15300: begin
paulson@15300: 
haftmann@40812: subsection {* Equivalence relations -- set version *}
paulson@15300: 
haftmann@40812: definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
haftmann@40812:   "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
paulson@15300: 
haftmann@40815: lemma equivI:
haftmann@40815:   "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
haftmann@40815:   by (simp add: equiv_def)
haftmann@40815: 
haftmann@40815: lemma equivE:
haftmann@40815:   assumes "equiv A r"
haftmann@40815:   obtains "refl_on A r" and "sym r" and "trans r"
haftmann@40815:   using assms by (simp add: equiv_def)
haftmann@40815: 
paulson@15300: text {*
paulson@15300:   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
paulson@15300:   r = r"}.
paulson@15300: 
paulson@15300:   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
paulson@15300: *}
paulson@15300: 
paulson@15300: lemma sym_trans_comp_subset:
paulson@15300:     "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
haftmann@46752:   by (unfold trans_def sym_def converse_unfold) blast
paulson@15300: 
nipkow@30198: lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"
nipkow@30198:   by (unfold refl_on_def) blast
paulson@15300: 
paulson@15300: lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
paulson@15300:   apply (unfold equiv_def)
paulson@15300:   apply clarify
paulson@15300:   apply (rule equalityI)
nipkow@30198:    apply (iprover intro: sym_trans_comp_subset refl_on_comp_subset)+
paulson@15300:   done
paulson@15300: 
paulson@15300: text {* Second half. *}
paulson@15300: 
paulson@15300: lemma comp_equivI:
paulson@15300:     "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
nipkow@30198:   apply (unfold equiv_def refl_on_def sym_def trans_def)
paulson@15300:   apply (erule equalityE)
paulson@15300:   apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
paulson@15300:    apply fast
paulson@15300:   apply fast
paulson@15300:   done
paulson@15300: 
paulson@15300: 
paulson@15300: subsection {* Equivalence classes *}
paulson@15300: 
paulson@15300: lemma equiv_class_subset:
paulson@15300:   "equiv A r ==> (a, b) \<in> r ==> r``{a} \<subseteq> r``{b}"
paulson@15300:   -- {* lemma for the next result *}
paulson@15300:   by (unfold equiv_def trans_def sym_def) blast
paulson@15300: 
paulson@15300: theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r``{a} = r``{b}"
paulson@15300:   apply (assumption | rule equalityI equiv_class_subset)+
paulson@15300:   apply (unfold equiv_def sym_def)
paulson@15300:   apply blast
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r``{a}"
nipkow@30198:   by (unfold equiv_def refl_on_def) blast
paulson@15300: 
paulson@15300: lemma subset_equiv_class:
paulson@15300:     "equiv A r ==> r``{b} \<subseteq> r``{a} ==> b \<in> A ==> (a,b) \<in> r"
paulson@15300:   -- {* lemma for the next result *}
nipkow@30198:   by (unfold equiv_def refl_on_def) blast
paulson@15300: 
paulson@15300: lemma eq_equiv_class:
paulson@15300:     "r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
nipkow@17589:   by (iprover intro: equalityD2 subset_equiv_class)
paulson@15300: 
paulson@15300: lemma equiv_class_nondisjoint:
paulson@15300:     "equiv A r ==> x \<in> (r``{a} \<inter> r``{b}) ==> (a, b) \<in> r"
paulson@15300:   by (unfold equiv_def trans_def sym_def) blast
paulson@15300: 
paulson@15300: lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
nipkow@30198:   by (unfold equiv_def refl_on_def) blast
paulson@15300: 
paulson@15300: theorem equiv_class_eq_iff:
paulson@15300:   "equiv A r ==> ((x, y) \<in> r) = (r``{x} = r``{y} & x \<in> A & y \<in> A)"
paulson@15300:   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
paulson@15300: 
paulson@15300: theorem eq_equiv_class_iff:
paulson@15300:   "equiv A r ==> x \<in> A ==> y \<in> A ==> (r``{x} = r``{y}) = ((x, y) \<in> r)"
paulson@15300:   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
paulson@15300: 
paulson@15300: 
paulson@15300: subsection {* Quotients *}
paulson@15300: 
haftmann@28229: definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90) where
haftmann@37767:   "A//r = (\<Union>x \<in> A. {r``{x}})"  -- {* set of equiv classes *}
paulson@15300: 
paulson@15300: lemma quotientI: "x \<in> A ==> r``{x} \<in> A//r"
paulson@15300:   by (unfold quotient_def) blast
paulson@15300: 
paulson@15300: lemma quotientE:
paulson@15300:   "X \<in> A//r ==> (!!x. X = r``{x} ==> x \<in> A ==> P) ==> P"
paulson@15300:   by (unfold quotient_def) blast
paulson@15300: 
paulson@15300: lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
nipkow@30198:   by (unfold equiv_def refl_on_def quotient_def) blast
paulson@15300: 
paulson@15300: lemma quotient_disj:
paulson@15300:   "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
paulson@15300:   apply (unfold quotient_def)
paulson@15300:   apply clarify
paulson@15300:   apply (rule equiv_class_eq)
paulson@15300:    apply assumption
paulson@15300:   apply (unfold equiv_def trans_def sym_def)
paulson@15300:   apply blast
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma quotient_eqI:
paulson@15300:   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y" 
paulson@15300:   apply (clarify elim!: quotientE)
paulson@15300:   apply (rule equiv_class_eq, assumption)
paulson@15300:   apply (unfold equiv_def sym_def trans_def, blast)
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma quotient_eq_iff:
paulson@15300:   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)" 
paulson@15300:   apply (rule iffI)  
paulson@15300:    prefer 2 apply (blast del: equalityI intro: quotient_eqI) 
paulson@15300:   apply (clarify elim!: quotientE)
paulson@15300:   apply (unfold equiv_def sym_def trans_def, blast)
paulson@15300:   done
paulson@15300: 
nipkow@18493: lemma eq_equiv_class_iff2:
nipkow@18493:   "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
nipkow@18493: by(simp add:quotient_def eq_equiv_class_iff)
nipkow@18493: 
paulson@15300: 
paulson@15300: lemma quotient_empty [simp]: "{}//r = {}"
paulson@15300: by(simp add: quotient_def)
paulson@15300: 
paulson@15300: lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
paulson@15300: by(simp add: quotient_def)
paulson@15300: 
paulson@15300: lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
paulson@15300: by(simp add: quotient_def)
paulson@15300: 
paulson@15300: 
nipkow@15302: lemma singleton_quotient: "{x}//r = {r `` {x}}"
nipkow@15302: by(simp add:quotient_def)
nipkow@15302: 
nipkow@15302: lemma quotient_diff1:
nipkow@15302:   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
nipkow@15302: apply(simp add:quotient_def inj_on_def)
nipkow@15302: apply blast
nipkow@15302: done
nipkow@15302: 
paulson@15300: subsection {* Defining unary operations upon equivalence classes *}
paulson@15300: 
paulson@15300: text{*A congruence-preserving function*}
haftmann@40816: 
haftmann@44278: definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where
haftmann@40817:   "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
haftmann@40816: 
haftmann@40816: lemma congruentI:
haftmann@40816:   "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
haftmann@40817:   by (auto simp add: congruent_def)
haftmann@40816: 
haftmann@40816: lemma congruentD:
haftmann@40816:   "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
haftmann@40817:   by (auto simp add: congruent_def)
paulson@15300: 
wenzelm@19363: abbreviation
wenzelm@21404:   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
wenzelm@21404:     (infixr "respects" 80) where
wenzelm@19363:   "f respects r == congruent r f"
paulson@15300: 
paulson@15300: 
paulson@15300: lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
paulson@15300:   -- {* lemma required to prove @{text UN_equiv_class} *}
paulson@15300:   by auto
paulson@15300: 
paulson@15300: lemma UN_equiv_class:
paulson@15300:   "equiv A r ==> f respects r ==> a \<in> A
paulson@15300:     ==> (\<Union>x \<in> r``{a}. f x) = f a"
paulson@15300:   -- {* Conversion rule *}
paulson@15300:   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
paulson@15300:   apply (unfold equiv_def congruent_def sym_def)
paulson@15300:   apply (blast del: equalityI)
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma UN_equiv_class_type:
paulson@15300:   "equiv A r ==> f respects r ==> X \<in> A//r ==>
paulson@15300:     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
paulson@15300:   apply (unfold quotient_def)
paulson@15300:   apply clarify
paulson@15300:   apply (subst UN_equiv_class)
paulson@15300:      apply auto
paulson@15300:   done
paulson@15300: 
paulson@15300: text {*
paulson@15300:   Sufficient conditions for injectiveness.  Could weaken premises!
paulson@15300:   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
paulson@15300:   A ==> f y \<in> B"}.
paulson@15300: *}
paulson@15300: 
paulson@15300: lemma UN_equiv_class_inject:
paulson@15300:   "equiv A r ==> f respects r ==>
paulson@15300:     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
paulson@15300:     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
paulson@15300:     ==> X = Y"
paulson@15300:   apply (unfold quotient_def)
paulson@15300:   apply clarify
paulson@15300:   apply (rule equiv_class_eq)
paulson@15300:    apply assumption
paulson@15300:   apply (subgoal_tac "f x = f xa")
paulson@15300:    apply blast
paulson@15300:   apply (erule box_equals)
paulson@15300:    apply (assumption | rule UN_equiv_class)+
paulson@15300:   done
paulson@15300: 
paulson@15300: 
paulson@15300: subsection {* Defining binary operations upon equivalence classes *}
paulson@15300: 
paulson@15300: text{*A congruence-preserving function of two arguments*}
haftmann@40817: 
haftmann@44278: definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
haftmann@40817:   "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
haftmann@40817: 
haftmann@40817: lemma congruent2I':
haftmann@40817:   assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
haftmann@40817:   shows "congruent2 r1 r2 f"
haftmann@40817:   using assms by (auto simp add: congruent2_def)
haftmann@40817: 
haftmann@40817: lemma congruent2D:
haftmann@40817:   "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
haftmann@40817:   using assms by (auto simp add: congruent2_def)
paulson@15300: 
paulson@15300: text{*Abbreviation for the common case where the relations are identical*}
nipkow@19979: abbreviation
wenzelm@21404:   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
wenzelm@21749:     (infixr "respects2" 80) where
nipkow@19979:   "f respects2 r == congruent2 r r f"
nipkow@19979: 
paulson@15300: 
paulson@15300: lemma congruent2_implies_congruent:
paulson@15300:     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
nipkow@30198:   by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast
paulson@15300: 
paulson@15300: lemma congruent2_implies_congruent_UN:
paulson@15300:   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
paulson@15300:     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
paulson@15300:   apply (unfold congruent_def)
paulson@15300:   apply clarify
paulson@15300:   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
paulson@15300:   apply (simp add: UN_equiv_class congruent2_implies_congruent)
nipkow@30198:   apply (unfold congruent2_def equiv_def refl_on_def)
paulson@15300:   apply (blast del: equalityI)
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma UN_equiv_class2:
paulson@15300:   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
paulson@15300:     ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
paulson@15300:   by (simp add: UN_equiv_class congruent2_implies_congruent
paulson@15300:     congruent2_implies_congruent_UN)
paulson@15300: 
paulson@15300: lemma UN_equiv_class_type2:
paulson@15300:   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
paulson@15300:     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
paulson@15300:     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
paulson@15300:     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
paulson@15300:   apply (unfold quotient_def)
paulson@15300:   apply clarify
paulson@15300:   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
paulson@15300:     congruent2_implies_congruent quotientI)
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma UN_UN_split_split_eq:
paulson@15300:   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
paulson@15300:     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
paulson@15300:   -- {* Allows a natural expression of binary operators, *}
paulson@15300:   -- {* without explicit calls to @{text split} *}
paulson@15300:   by auto
paulson@15300: 
paulson@15300: lemma congruent2I:
paulson@15300:   "equiv A1 r1 ==> equiv A2 r2
paulson@15300:     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
paulson@15300:     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
paulson@15300:     ==> congruent2 r1 r2 f"
paulson@15300:   -- {* Suggested by John Harrison -- the two subproofs may be *}
paulson@15300:   -- {* \emph{much} simpler than the direct proof. *}
nipkow@30198:   apply (unfold congruent2_def equiv_def refl_on_def)
paulson@15300:   apply clarify
paulson@15300:   apply (blast intro: trans)
paulson@15300:   done
paulson@15300: 
paulson@15300: lemma congruent2_commuteI:
paulson@15300:   assumes equivA: "equiv A r"
paulson@15300:     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
paulson@15300:     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
paulson@15300:   shows "f respects2 r"
paulson@15300:   apply (rule congruent2I [OF equivA equivA])
paulson@15300:    apply (rule commute [THEN trans])
paulson@15300:      apply (rule_tac [3] commute [THEN trans, symmetric])
paulson@15300:        apply (rule_tac [5] sym)
haftmann@25482:        apply (rule congt | assumption |
paulson@15300:          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
paulson@15300:   done
paulson@15300: 
haftmann@24728: 
haftmann@24728: subsection {* Quotients and finiteness *}
haftmann@24728: 
wenzelm@40945: text {*Suggested by Florian Kammüller*}
haftmann@24728: 
haftmann@24728: lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
haftmann@24728:   -- {* recall @{thm equiv_type} *}
haftmann@24728:   apply (rule finite_subset)
haftmann@24728:    apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
haftmann@24728:   apply (unfold quotient_def)
haftmann@24728:   apply blast
haftmann@24728:   done
haftmann@24728: 
haftmann@24728: lemma finite_equiv_class:
haftmann@24728:   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
haftmann@24728:   apply (unfold quotient_def)
haftmann@24728:   apply (rule finite_subset)
haftmann@24728:    prefer 2 apply assumption
haftmann@24728:   apply blast
haftmann@24728:   done
haftmann@24728: 
haftmann@24728: lemma equiv_imp_dvd_card:
haftmann@24728:   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
haftmann@24728:     ==> k dvd card A"
berghofe@26791:   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
haftmann@24728:    apply assumption
haftmann@24728:   apply (rule dvd_partition)
haftmann@24728:      prefer 3 apply (blast dest: quotient_disj)
haftmann@24728:     apply (simp_all add: Union_quotient equiv_type)
haftmann@24728:   done
haftmann@24728: 
haftmann@24728: lemma card_quotient_disjoint:
haftmann@24728:  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
haftmann@24728: apply(simp add:quotient_def)
haftmann@24728: apply(subst card_UN_disjoint)
haftmann@24728:    apply assumption
haftmann@24728:   apply simp
nipkow@44890:  apply(fastforce simp add:inj_on_def)
huffman@35216: apply simp
haftmann@24728: done
haftmann@24728: 
haftmann@40812: 
haftmann@40812: subsection {* Equivalence relations -- predicate version *}
haftmann@40812: 
haftmann@40812: text {* Partial equivalences *}
haftmann@40812: 
haftmann@40812: definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40812:   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
haftmann@40812:     -- {* John-Harrison-style characterization *}
haftmann@40812: 
haftmann@40812: lemma part_equivpI:
haftmann@40812:   "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
haftmann@45969:   by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
haftmann@40812: 
haftmann@40812: lemma part_equivpE:
haftmann@40812:   assumes "part_equivp R"
haftmann@40812:   obtains x where "R x x" and "symp R" and "transp R"
haftmann@40812: proof -
haftmann@40812:   from assms have 1: "\<exists>x. R x x"
haftmann@40812:     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
haftmann@40812:     by (unfold part_equivp_def) blast+
haftmann@40812:   from 1 obtain x where "R x x" ..
haftmann@40812:   moreover have "symp R"
haftmann@40812:   proof (rule sympI)
haftmann@40812:     fix x y
haftmann@40812:     assume "R x y"
haftmann@40812:     with 2 [of x y] show "R y x" by auto
haftmann@40812:   qed
haftmann@40812:   moreover have "transp R"
haftmann@40812:   proof (rule transpI)
haftmann@40812:     fix x y z
haftmann@40812:     assume "R x y" and "R y z"
haftmann@40812:     with 2 [of x y] 2 [of y z] show "R x z" by auto
haftmann@40812:   qed
haftmann@40812:   ultimately show thesis by (rule that)
haftmann@40812: qed
haftmann@40812: 
haftmann@40812: lemma part_equivp_refl_symp_transp:
haftmann@40812:   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
haftmann@40812:   by (auto intro: part_equivpI elim: part_equivpE)
haftmann@40812: 
haftmann@40812: lemma part_equivp_symp:
haftmann@40812:   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
haftmann@40812:   by (erule part_equivpE, erule sympE)
haftmann@40812: 
haftmann@40812: lemma part_equivp_transp:
haftmann@40812:   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
haftmann@40812:   by (erule part_equivpE, erule transpE)
haftmann@40812: 
haftmann@40812: lemma part_equivp_typedef:
kaliszyk@44204:   "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
kaliszyk@44204:   by (auto elim: part_equivpE)
haftmann@40812: 
haftmann@40812: 
haftmann@40812: text {* Total equivalences *}
haftmann@40812: 
haftmann@40812: definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40812:   "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}
haftmann@40812: 
haftmann@40812: lemma equivpI:
haftmann@40812:   "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
haftmann@45969:   by (auto elim: reflpE sympE transpE simp add: equivp_def)
haftmann@40812: 
haftmann@40812: lemma equivpE:
haftmann@40812:   assumes "equivp R"
haftmann@40812:   obtains "reflp R" and "symp R" and "transp R"
haftmann@40812:   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
haftmann@40812: 
haftmann@40812: lemma equivp_implies_part_equivp:
haftmann@40812:   "equivp R \<Longrightarrow> part_equivp R"
haftmann@40812:   by (auto intro: part_equivpI elim: equivpE reflpE)
haftmann@40812: 
haftmann@40812: lemma equivp_equiv:
haftmann@40812:   "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
haftmann@46752:   by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
haftmann@40812: 
haftmann@40812: lemma equivp_reflp_symp_transp:
haftmann@40812:   shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
haftmann@40812:   by (auto intro: equivpI elim: equivpE)
haftmann@40812: 
haftmann@40812: lemma identity_equivp:
haftmann@40812:   "equivp (op =)"
haftmann@40812:   by (auto intro: equivpI reflpI sympI transpI)
haftmann@40812: 
haftmann@40812: lemma equivp_reflp:
haftmann@40812:   "equivp R \<Longrightarrow> R x x"
haftmann@40812:   by (erule equivpE, erule reflpE)
haftmann@40812: 
haftmann@40812: lemma equivp_symp:
haftmann@40812:   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
haftmann@40812:   by (erule equivpE, erule sympE)
haftmann@40812: 
haftmann@40812: lemma equivp_transp:
haftmann@40812:   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
haftmann@40812:   by (erule equivpE, erule transpE)
haftmann@40812: 
paulson@15300: end
haftmann@46752: